One-loop QCD contributions to differential cross-sections for Higgs production at N$^3$LO

We present one-loop contributions to the fully differential Higgs boson gluon-fusion cross-section for Higgs production via gluon fusion. Our results constitute a necessary ingredient of a complete N$^3$LO determination of the cross-section. We perform our computation using a subtraction method for the treatment of soft and collinear singularities. We identify the infrared divergent parts in terms of universal splitting and eikonal functions, and demonstrate how phase-space integrations yield poles (up to $1/\epsilon^6$ ) in the dimensional regulator $\epsilon=(4-d)/2$. We compute the coefficients of the $\epsilon$ expansion, including the finite part numerically. As a demonstration of our numerical implementation, we present the corrections at N$^3$LO due to one-loop amplitudes in the rapidity and transverse momentum of the Higgs boson.